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Digital Image Processing

D. Sundararajan

Chapter 10

Segmentation - all with Video Answers

Educators

Chapter Questions

Problem 1

Using the averaging method, find the threshold of the $4 \times 4$-bit image. Let the initial value of the threshold be the average of the gray levels of the image. The iteration stops when the difference between two consecutive threshold values becomes less than 0.5 .
$$
\left[\begin{array}{rrrr}
140 & 10 & 5 & 6 \\
74 & 2 & 7 & 6 \\
21 & 5 & 6 & 5 \\
2 & 6 & 5 & 5
\end{array}\right]
$$

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Problem 2

Using the averaging method, find the threshold of the $4 \times 4$ 8-bit image. Let the initial value of the threshold be the average of the gray levels of the image. The iteration stops when the difference between two consecutive threshold values becomes less than 0.5 .
$$
\left[\begin{array}{rrrr}
191 & 102 & 1 & 7 \\
182 & 45 & 2 & 6 \\
140 & 10 & 5 & 6 \\
74 & 2 & 7 & 6
\end{array}\right]
$$

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Problem 3

Using the averaging method, find the threshold of the $4 \times 4$-bit image. Let the initial value of the threshold be the average of the gray levels of the image. The iteration stops when the difference between two consecutive threshold values becomes less than 0.5 .
$$
\left[\begin{array}{rrrr}
184 & 188 & 72 & 2 \\
188 & 163 & 22 & 5 \\
191 & 102 & 1 & 7 \\
182 & 45 & 2 & 6
\end{array}\right]
$$

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Problem 4

Using Otsu's method, find the threshold of the $4 \times 4$ 3-bit image. Find the separability index.
$$
\left[\begin{array}{llll}
5 & 2 & 6 & 5 \\
2 & 5 & 6 & 6 \\
2 & 7 & 6 & 6 \\
5 & 6 & 5 & 5
\end{array}\right]
$$

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Problem 5

Using Otsu's method, find the threshold of the $4 \times 4$ 3-bit image. Find the separability index.
$$
\left[\begin{array}{llll}
4 & 0 & 2 & 6 \\
3 & 6 & 5 & 6 \\
6 & 1 & 7 & 6 \\
5 & 2 & 6 & 5
\end{array}\right]
$$

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Problem 6

Using Otsu's method, find the threshold of the $4 \times 4$ 3-bit image. Find the separability index.
$$
\left[\begin{array}{llll}
5 & 6 & 5 & 5 \\
6 & 5 & 5 & 6 \\
7 & 6 & 4 & 5 \\
5 & 5 & 5 & 5
\end{array}\right]
$$

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Problem 7

Consider the $8 \times 8$ image. The seed pixel is shown in boldface. Use the 4 connectivity to segment the image so that the region is to be made of pixels with gray levels less than or equal to 176 .
$$
\left[\begin{array}{rrrrrrrr}
255 & 207 & 73 & 38 & 42 & 43 & 42 & 40 \\
255 & 255 & 205 & 46 & 43 & 45 & 43 & 42 \\
255 & 255 & 253 & 84 & 36 & 43 & 46 & 43 \\
255 & 255 & 255 & 126 & 37 & 45 & 4644 \\
255 & 255 & 255 & 169 & 35 & 45 & 46 & 45 \\
255 & 255 & 255 & 222 & 54 & 37 & 42 & 42 \\
255 & 255 & 255 & 242 & 120 & 28 & 3939 \\
255 & 255 & 255 & 240 & 212 & 92 & 3342
\end{array}\right]
$$

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Problem 8

Consider the $8 \times 8$ image. The seed pixel is shown in boldface. Use the $4-$ connectivity to segment the image so that the region is to be made of pixels with gray levels less than or equal to 76 .
$$
\left[\begin{array}{rrrrrrrr}
172 & 157 & 115 & 62 & 4 & 4 & 4 & 46 \\
163 & 165 & 118 & 83 & 13 & 6 & 9 & 46 \\
138 & 185 & 128 & 71 & 90 & 24 & 19 & 30 \\
121 & 184 & 126 & 83 & 78 & 51 & 50 & 51 \\
156 & 185 & 136 & 74 & 40 & 42 & 44 & 53 \\
160 & 175 & 121 & 63 & 77 & 81 & 65 & 71 \\
178 & 170 & 128 & 88 & 73 & 61 & 59 & 59 \\
176 & 163 & 133 & 97 & 88 & 35 & 26 & 27
\end{array}\right]
$$

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Problem 9

Consider the $8 \times 8$ image. The seed pixel is shown in boldface. Use the 4 connectivity to segment the image so that the region is to be made of pixels with gray levels less than or equal to 56 .
$$
\left[\begin{array}{llllllll}
65 & 62 & 65 & 61 & 59 & 57 & 56 & 54 \\
58 & 62 & 64 & 62 & 58 & 58 & 54 & 52 \\
71 & 71 & 64 & 63 & 58 & 55 & 54 & 54 \\
71 & 71 & 70 & 70 & 62 & 57 & 54 & 55 \\
71 & 70 & 68 & 69 & 66 & 62 & 59 & 58 \\
69 & 66 & 67 & 67 & 62 & 61 & 60 & 58 \\
65 & 62 & 63 & 65 & 64 & 61 & 60 & 55 \\
66 & 61 & 61 & 61 & 62 & 62 & 60 & 57
\end{array}\right]
$$

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Problem 10

Consider the $8 \times 8$ image. Use the 8 -connectivity to segment, using the split and merge algorithm, the image so that the region is to be made of pixels with gray levels less than or equal to 45 .
$$
\left[\begin{array}{llllllll}
59 & 50 & 40 & 29 & 22 & 20 & 20 & 21 \\
55 & 48 & 39 & 28 & 22 & 20 & 20 & 20 \\
53 & 47 & 38 & 28 & 22 & 20 & 20 & 22 \\
54 & 47 & 38 & 28 & 22 & 20 & 20 & 23 \\
57 & 49 & 40 & 29 & 23 & 21 & 21 & 22 \\
62 & 54 & 44 & 32 & 25 & 22 & 22 & 20 \\
69 & 60 & 49 & 36 & 28 & 23 & 22 & 20 \\
78 & 68 & 55 & 41 & 31 & 24 & 22 & 22
\end{array}\right]
$$

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Problem 11

Consider the $8 \times 8$ image. Use the 8 -connectivity to segment, using the split and merge algorithm, the image so that the region is to be made of pixels with gray levels less than or equal to 240 .
$$
\left[\begin{array}{llllllll}
248 & 247 & 242 & 238 & 236 & 235 & 235 & 234 \\
248 & 247 & 243 & 241 & 238 & 235 & 235 & 234 \\
249 & 248 & 244 & 242 & 239 & 235 & 235 & 235 \\
249 & 248 & 246 & 244 & 240 & 235 & 235 & 235 \\
249 & 248 & 247 & 245 & 241 & 236 & 236 & 235 \\
247 & 247 & 246 & 244 & 241 & 236 & 236 & 236 \\
244 & 245 & 244 & 243 & 241 & 236 & 236 & 236 \\
241 & 243 & 242 & 241 & 239 & 236 & 236 & 236
\end{array}\right]
$$

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Problem 12

Consider the $8 \times 8$ image. Use the 8 -connectivity to segment, using the split and merge algorithm, the image so that the region is to be made of pixels with gray levels less than or equal to 240 .
$$
\left[\begin{array}{llllllll}
239 & 240 & 240 & 240 & 241 & 243 & 238 & 231 \\
239 & 240 & 240 & 240 & 241 & 242 & 239 & 234 \\
239 & 240 & 240 & 240 & 240 & 240 & 240 & 238 \\
239 & 240 & 240 & 240 & 239 & 238 & 240 & 241 \\
239 & 240 & 240 & 240 & 239 & 239 & 242 & 244 \\
237 & 238 & 239 & 240 & 241 & 245 & 245 & 245 \\
236 & 238 & 239 & 240 & 241 & 245 & 246 & 245 \\
236 & 238 & 239 & 240 & 241 & 245 & 246 & 245
\end{array}\right]
$$

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Problem 13

Find the seed points of the regions by clustering and finding the centroids. Initial seeds are $\{2,2\}$ and $\{2,3\}$.
$$
\left[\begin{array}{llllllll}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]
$$

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Problem 14

Find the seed points of the regions by clustering and finding the centroids. Initial seeds are $\{2,2\}$ and $\{2,3\}$.
$$
\left[\begin{array}{llllllll}
0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1
\end{array}\right]
$$

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Problem 15

Find the seed points of the regions by clustering and finding the centroids. Initial seeds are $\{2,2\}$ and $\{2,3\}$.
$$
\left[\begin{array}{llllllll}
0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 0
\end{array}\right]
$$

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01:18

Problem 16

Using the distance transform with masks
$$
h_f(m, n)=\left[\begin{array}{rrr}
\infty & 1 & \infty \\
1 & 0 & \infty \\
\infty & \infty & \infty
\end{array}\right] \text { and } h_b(m, n)=\left[\begin{array}{ccc}
\infty & \infty & \infty \\
\infty & 0 & 1 \\
\infty & 1 & \infty
\end{array}\right]
$$
find the distance of the pixels of the $8 \times 8$ image.
$$
\left[\begin{array}{llllllll}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]
$$

Arun Bana
Arun Bana
Numerade Educator
01:18

Problem 17

Using the distance transform with masks
$$
h_f(m, n)=\left[\begin{array}{rrr}
4 & 3 & 4 \\
3 & 0 & \infty \\
\infty & \infty & \infty
\end{array}\right] \text { and } h_b(m, n)=\left[\begin{array}{ccc}
\infty & \infty & \infty \\
\infty & 0 & 3 \\
4 & 3 & 4
\end{array}\right]
$$
find the distance of the pixels of the $8 \times 8$ image.
$$
\left[\begin{array}{llllllll}
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 0 & 0
\end{array}\right]
$$

Arun Bana
Arun Bana
Numerade Educator
01:18

Problem 18

Using the distance transform with masks
$$
h_f(m, n)=\left[\begin{array}{rrrrr}
\infty & 11 & \infty & 11 & \infty \\
11 & 7 & 5 & 7 & 11 \\
\infty & 5 & 0 & \infty & \infty \\
\infty & \infty & \infty & \infty & \infty \\
\infty & \infty & \infty & \infty & \infty
\end{array}\right] \text { and } h_b(m, n)=\left[\begin{array}{rrrrr}
\infty & \infty & \infty & \infty & \infty \\
\infty & \infty & \infty & \infty & \infty \\
\infty & \infty & 0 & 5 & \infty \\
11 & 7 & 5 & 7 & 11 \\
\infty & 11 & \infty & 11 & \infty
\end{array}\right]
$$
find the distance of the pixels of the $8 \times 8$ image.
$$
\left[\begin{array}{llllllll}
0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{array}\right]
$$

Arun Bana
Arun Bana
Numerade Educator